Finite Group Databasebeta

← #399×D127 #30641976

Subgroups — #399×D127 #30641976

Order 16256.

908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D127C213yesmaximal, complemented
2C2×C22⋊C4×D127C213yesmaximal
3#22549813C211yesmaximal, characteristic, complemented
4#22549812C213yesmaximal, complemented
5#22549811C213yesmaximal, complemented
6#95×D127C211yesmaximal, characteristic, complemented
7#399×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C22×C4×D127C2213yes—
10C22×C4×D127C2213yes—
11C2×C4⋊Dic127C2213yescomplemented
12C2×Dic127⋊C4C2216yescomplemented
13C4⋊C4×D127—212no—
14C2×C22⋊Dic127C2213yes—
15C2×(C2×D127)⋊C4C2213yes—
16C2×(C2×D127)⋊C4C2213yes—
17C22⋊C4×D127—212no—
18C127⋊#95C2213yescomplemented
19C2×C4⋊C4×C127C2213yescomplemented
20C2×C22⋊C4×C127C2213yes—
21#95×C127C2211yescharacteristic, complemented
22C23×D127—26no—
23C23×D127—26no—
24C23×D127Q814yes—
25C23×D127C2311yescharacteristic
26C23×D127C2311yescharacteristic
27C22×Dic127C2313yes—
28C22×Dic127C2313yes—
29C2×C4×D127—212no—
30C2×C4×D127—26no—
31C2×C4×D127—212no—
32C2×C4×D127D4112yes—
33C4⋊Dic127—26no—
34Dic127⋊C4—212no—
35C22⋊Dic127—26no—
36(C2×D127)⋊C4—26no—
37(C2×D127)⋊C4—26no—
38C24×C127C2311yescharacteristic
39C22×C4×C127C2313yes—
40C22×C4×C127C2313yes—
41C4⋊C4×C127—26no—
42C22⋊C4×C127—26no—
43C22×D127—23no—
44C22×D127—23no—
45C22×D127—224no—
46C22×D127—412no—
47C22×D127—44no—
48C22×D127C2×D416yes—
49C22×D127C2×Q812yes—
50C22×D127C2×Q812yes—
51C22×D127#4116yes—
52C2×Dic127—26no—
53C2×Dic127—23no—
54C2×Dic127—26no—
55C2×Dic127C2×D416yes—
56C4×D127—224no—
57C4×D127—412no—
58C23×C127—23no—
59C23×C127—23no—
60C23×C127C2×Q812yes—
61C23×C127C2411yescharacteristic
62C2×C4×C127—26no—
63C2×C4×C127—23no—
64C2×C4×C127—26no—
65C2×C4×C127C2×D416yes—
66C22×C127—26no—
67C22×C127—43no—
68C22×C127—41no—
69C22×C127C22×D413yes—
70C22×C127C22×Q811yescharacteristic
71C22×C127C2×#4113yes—
72C2×D127—26no—
73C2×D127—43no—
74C2×D127—41no—
75C2×D127—416no—
76C2×D127C22⋊Q8112yes—
77C2×D127#123112yes—
78C2×D127#12714yes—
79C4×C127—26no—
80C4×C127—43no—
81Dic127—26no—
82Dic127—43no—
83C2×C127—42no—
84C2×C127C2×C22⋊Q813yes—
85C2×C127C2×#12313yes—
86C2×C127C2×#12711yescharacteristic
87D127—42no—
88D127#39918yesdirect product factor
89C2×#399—1271nomaximal, Sylow
90C127C2×#39911yescharacteristic, complemented, Sylow
91C22×C4⋊C4—1273no—
92C22×C22⋊C4—1273no—
93#399—1273no—
94#399—1273no—
95#399—1271no—
96#399D12711yescharacteristic, direct product factor
97C2×#95—1271no—
98C25—1271no—
99C23×C4—1273no—
100C23×C4—1273no—
101C2×C4⋊C4—1276no—
102C2×C4⋊C4—1273no—
103C2×C4⋊C4—25412no—
104C2×C4⋊C4C2×D12713yescomplemented
105C2×C22⋊C4—1273no—
106C2×C22⋊C4—1273no—
107C2×C22⋊C4—1273no—
108C2×C22⋊C4—25412no—
109C2×C22⋊C4C2×D12713yes—
110#95—1273no—
111#95C2×D12711yescharacteristic, complemented
112C24—1271no—
113C24—1274no—
114C24—1271no—
115C24—2546no—
116C24—2546no—
117C24C22×D12711yescharacteristic
118C22×C4—1273no—
119C22×C4—12712no—
120C22×C4—1273no—
121C22×C4—2546no—
122C22×C4—25412no—
123C22×C4—25412no—
124C22×C4C22×D12713yes—
125C22×C4C22×D12713yes—
126C4⋊C4—26no—
127C4⋊C4—25412no—
128C4⋊C4—2546no—
129C22⋊C4—26no—
130C22⋊C4—2546no—
131C22⋊C4—2546no—
132C22⋊C4—2546no—
133C23—23no—
134C23—23no—
135C23—1276no—
136C23—1272no—
137C23—1276no—
138C23—1272no—
139C23—50812no—
140C23—5084no—
141C23—25424no—
142C23—2543no—
143C23—2543no—
144C23Q8×D12712yes—
145C23C23×D12711yescharacteristic
146C2×C4—26no—
147C2×C4—23no—
148C2×C4—26no—
149C2×C4—1276no—
150C2×C4—50812no—
151C2×C4—25424no—
152C2×C4—2543no—
153C2×C4—2546no—
154C2×C4—2546no—
155C2×C4D4×D12716yes—
156C22—26no—
157C22—43no—
158C22—41no—
159C22—12712no—
160C22—12712no—
161C22—1274no—
162C22—50816no—
163C22—5083no—
164C22—5081no—
165C22—2546no—
166C22C2×D4×D12713yes—
167C22C2×Q8×D12711yescharacteristic
168C22#41×D12713yes—
169C4—26no—
170C4—43no—
171C4—5083no—
172C4—2546no—
173C2—42no—
174C2—1278no—
175C2—5082no—
176C2C22⋊Q8×D12713yes—
177C2#123×D12713yes—
178C2#127×D12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#399×C127C2
prime radicalC24×D127C22
hypercenter#399D127
hypocenterC127C2×#399
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #399

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C4⋊C4×D127C2semidirect
3. #22549813C2semidirect
4. #22549812C2semidirect
5. #22549811C2semidirect
6. #95×D127C2semidirect
7. #399×C127C2semidirect
11. C2×C4⋊Dic127C22semidirect
12. C2×Dic127⋊C4C22semidirect
18. C127⋊#95C22semidirect
19. C2×C4⋊C4×C127C22semidirect
21. #95×C127C22semidirect
88. D127#399direct (N × H)
90. C127C2×#399semidirect
96. #399D127direct (N × H)
104. C2×C4⋊C4C2×D127semidirect
111. #95C2×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4⋊C4×D1272
C2×C22⋊C4×D1272
#225498132
#225498122
#225498112
#95×D1272
#399×C1272
C2×#399127